Finally, the restriction of the logarithmic kernel \(-\log \,|x-y|\)  to a closed disc in \(\mathbb {R}^2\)  of radius \({}<1\)  is strictly positive definite and fulfills Frostman's maximum principle [1]}; hence, it is perfect [2]}, and satisfies the principle of positivity of mass (cf. Theorem REF  and Remark REF ). However, the domination principle then fails in general; it does hold only in a weaker sense where the measures \(\mu ,\nu \)  involved in the above-quoted definition (see Sect. REF ) meet the additional requirement that \(\nu (\mathbb {R}^2)\leqslant \mu (\mathbb {R}^2)\) , cf. [3]}.
